There are some equivalents (ie., if A=B and B=C, then A=C) in the Gospel when it comes to this subject. The ministry of the Second Comforter is to bring those to whom He ministers to the Father, and have them accepted by Him. This means that the Father accepts them as a member of the Heavenly Family, or in other words, promises them exaltation.
Since the end of that ministry is to have the person accepted by the Father as a son or daughter of God, then an equivalency can be drawn between the final outcome and the Second Comforter. This is what is done in D&C 88: 3-5. Joseph Smith did something similar in a statement he made in which he put the voice declaring a person's exaltation first, and the visit of Christ and the Father with that person second. You can read about it in the Words of Joseph Smith, pages 3-6, but the most relevant excerpt is found below:
Now what is this other Comforter? It is no more or less than the Lord Jesus Christ himself & this is the sum & substance of the whole matter, that when any man obtains this last Comforter he will have the personage of Jesus Christ to attend him or appear unto him from time to time. & even he will manifest the Father unto him & they will take up their abode with him, & the visions of the heavens will be opened unto him & the Lord will teach him face to face & he may have a perfect knowledge of the mysteries of the kingdom of God, & this is the state & place the Ancient Saints arrived at when they had such glorious vision Isaiah, Ezekiel, John upon the Isle of Patmos, St Paul in the third heavens, & all the Saints who held communion with the general Assembly & Church of the First Born &c.
Since the white stone and new name mentioned in D&C 130: 10-11 are referring to the state of exaltation and inheritance, and since the promise which the Second Comforter (Christ) is working to obtain for those to whom He ministers is the promise of exaltation, that equivalency may also be made. The difference as I see it is that those described in the verses in D&C 130 are in a future state, in which they have actually inherited the condition of exaltation, have entered into the Celestial Kingdom to dwell there and possess the white stone on which their new name is written; whereas the promises Joseph speaks of in the quote above and the promises in D&C 88 are given to a mortal and are to be realized fully in the future.
Now the promise of the Lord is reality itself. What He says will happen. His Word becomes the law of the universe. (See D&C 1: 38.) Therefore when viewed with the eyes of faith, the Word is the reality, and the inheritance is immediate for those with faith. This is the reason why Joseph said when a man receives "this last Comforter he will have the personage of Jesus Christ to attend him or appear unto him from time to time. & even he will manifest the Father unto him & they will take up their abode with him, & the visions of the heavens will be opened unto him & the Lord will teach him face to face & he may have a perfect knowledge of the mysteries of the kingdom of God[.]"
Finally, since the mortal who receives these things is already in company with the Lord and the Father, they are already occasional visitors in a Celestial Kingdom although they are still here in mortality, required to endure to the end, suffer death and then await resurrection. Despite this, they are celestial and their lives are punctuated by contact with celestial beings from time to time, as the Lord determines is appropriate or necessary.